Modelling epidemics on networks with...
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AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
Modelling epidemics on networks with cliques
Anlin ZhangMentor: Professor Laura Schaposnik
Seventh Annual PRIMES Conference
May 20, 2017
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 1 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
THE PLAN
1 An introduction to modeling epidemics.
2 Modeling epidemics on regular tree graphs.
3 Modeling epidemics on networks with cliques.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 2 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
AN INTRODUCTION TO MODELING EPIDEMICSHISTORY
1 1920s: Reed-Frost model investigated the number of individuals infectedand immune for certain periods of time
2 1927: Kermack and McKendrick’s SIR model with 3 states Susceptible,Infectious and Removed
3 2005: Keeling and Eames’s contact networks
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 3 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
AN INTRODUCTION TO MODELING EPIDEMICSMODELING THROUGH GRAPHS
Each vertex denotes an individual.
Edges denote possible paths of infection.
Figure: In this network, pink vertices represent females, blue vertices representmales, and stars represent the origins of infection.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 4 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON REGULAR TREE GRAPHS
DefinitionA tree is an undirected graph in which any two vertices are connected byexactly one path, and a rooted tree graph is a tree in which one vertex hasbeen designated the root. A regular tree graph is a tree graph in which everynonterminal vertex has the same number of edges connected to it. The heightλ is the largest distance between any node and root, and the degree d is thenumber of nodes connected to the root by an edge.
Figure: An example of a tree graph of degree d = 3 and height λ = 2.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 5 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON REGULAR TREE GRAPHSMODELING EPIDEMICS ON REGULAR TREE GRAPHS
IONTW PLATFORM
Figure: From [Seibold-Callender]
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 6 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON REGULAR TREE GRAPHSMODELING EPIDEMICS ON REGULAR TREE GRAPHS
RELATIONS BETWEEN PARAMETERS
Proposition (Seibold-Callender)Let N denote the number of nodes in a regular tree graph. Then
N =
λ∑i=0
di.
Figure: From [Seibold-Callender]
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 7 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON REGULAR TREE GRAPHS
DefinitionFor each node, Pinf is the probability of infection, while Prec is the probabilityof recovery.
DefinitionWe denote by δ the number of edges in the outbreak duration. δλ is theexpected average outbreak duration, given by
∑λ+1i=1 i · Pi, where Pi is the
probability the duration spans i edges.
Duration when Prec = 1 and Pinf = 1
I λ+ 1 ≤ δ ≤ 2λ+ 1
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 8 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON REGULAR TREE GRAPHS
Consider Pinf 6= 1. A specific node getting infected has probability Pinf . Then,
Both adjacent nodes remain uninfected with probability (1− Pinf )2.
One becomes infected with probability 2Pinf · (1− Pinf ).
Both become infected with probability P2inf .
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 9 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
Theorem (Seibold-Callender)Let Pi denote the probability that the outbreak duration is i. Then
P1 = (1− Pinf )d
P2 =
d∑r=1
(dr
)Pr
inf (1− Pinf )d−r(1− Pinf )
d·r
...
Pλ+1 = 1− (P1 + P2 + · · ·+ Pλ).
(Recall λ is the maximum distance, in edges, a node can exist from the root)
Question: How can we model outbreaks in more general types of graphs, toaccount for symmetries appearing, for example, from family structures?
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 10 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON REGULAR TREE GRAPHSHOW TO EXTEND THESE RESULTS TO BETTER ENCOMPASS REAL NETWORKS?
Irregular tree graphs
Additional variables such as age, weight, and race
Cliques in graphs
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 11 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON NETWORKS WITH CLIQUES
DefinitionA clique is a subset S of a graph such that all pairs of vertices in S areconnected by an edge. A clique with c vertices is called a c-clique. Amaximum clique is the clique of largest size in a graph. The clique number,denoted as ω(G), is the number of vertices in a maximum clique of G.
Figure: Clique number ω(G) = 4.
Question: How can we bound the clique number to understand the durationof outbreaks in terms of Prec and Pinf in this new setting?
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 12 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON NETWORKS WITH CLIQUESINDUCED CLIQUES
Theorem (Nikiforov)If G does not contain a complete subgraph of k + 1 vertices, then
λ ≤√
2(k − 1) · ek
,
where e denotes the number of edges.
From there, we can deduce:
PropositionThe number ω(G) is bounded by
1
1− λ2
2e
≤ ω(G).
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 13 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON NETWORKS WITH CLIQUESREGULAR TREE GRAPHS WITH INDUCED CLIQUES
Theorem (Turan)If G does not contain a complete subgraph of k + 1 vertices, then
e <k − 1
2k· n2,
where n is the number of vertices of G.
From there, we can deduce:
PropositionIn terms of n, we bound ω(G) by
nn− λ
≤ ω(G).
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 14 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON NETWORKS WITH CLIQUESROOT PLACEMENT
Question: How does the origin of infection affect outbreak duration?
Case 1: When the root is outside all cliques
Case 2: When the root is in a clique and connected to the "body" of G
Case 3: When the root is a vertex inside a clique
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 15 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
EPIDEMICS ON NETWORKS WITH CLIQUESSAFE ZONES
Question: How can one remove a given number of links or nodes of anetwork so that an outbreak is contained within a subset of the graph?
DefinitionThe safe zone of a graph is defined as a subset of vertices that cannot beinfected, and makes up one connected component. The number of vertices inthe safe zone shall be denoted by sz.
We remove edges or vertices following the Optimal Link Removal (OLR)strategy of [Yang et al.].
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 16 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
SUMMARY
We have studied epidemic outbreaks modeled on graphs:
(?) We have studied [Seibold-Callender] approach for regular tree graphs.
(?) We have considered graphs with cliques, and bounded the clique numberto understand the duraiton of outbreaks in terms of Prec and Pinf in thisnew setting.
(?) We have defined Safe zones and begun the study of their sz number.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 17 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
FURTHER WORK
In what remains of the year, we will do the following:
(?) Bound duration depending on root placement in networks with cliques.
(?) Expand to include multiple roots of infection.
(?) Generalize when the removal of a vertex or edge forms a safe zone.
(?) Determine how to maximize sz.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 18 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
ACKNOWLEDGEMENTSI SINCERELY THANK...
My mentor Professor Schaposnik for her invaluable suggestions andguidance.
Dr. Tanya Khovanova for her help and advice.
MIT PRIMES-USA program for this research opportunity.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 19 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
MAIN BIBLIOGRAPHY
Nikiforov, V. "Some Inequalities for the Largest Eigenvalue of a Graph."Combinatorics, Probability and Computing, 11(2), 179-189.
TurÃan, Paul (1941). "On an extremal problem in graph theory",Matematikai Ãl’s Fizikai Lapok (in Hungarian), 48: 436-452.
Seibold, Claire, and Hannah L. Callender. "Modeling Epidemics on aRegular Tree Graph." Letters in Biomathematics, 30 May 2016.
Yang, Xiaofan, Pengdeng Li, Lu-Xing Yang, and Yingbo Wu. "Reducingthe Spectral Radius of a Torus Network by Link Removal." PLOS ONE.Public Library of Science, 12 May 2016.
Kermack, W. O. and McKendrick, A. G. "A Contribution to theMathematical Theory of Epidemics." Proc. Roy. Soc. Lond. A 115,700-721, 1927.
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 20 / 21
AN INTRODUCTION TO MODELING EPIDEMICS Epidemics on regular tree graphs Epidemics on networks with cliques FURTHER THOUGHTS
Thank you for listening!Any questions?
Anlin Zhang Mentor: Professor Laura Schaposnik (School)Modelling epidemics on networks with cliques May 20, 2017 21 / 21